403 – Homework 2

This new set is due Monday, March 11, at the beginning of lecture.

  • Write a program that, given a square matrix A, computes approximations to its eigenvalues using the QR-algorithm. Ideally, the user can decide the dimensions of the matrix and, more importantly, the error within which the approximations will be found. Apply your method to a 4\times 4 matrix, and check the number of iterations the process requires.

Please turn in: The code (best if you email it to me), a write up explaining what your code does, the matrix you applied the method to, and the result. For this, you can work in groups of two or three. In case you cannot find anybody to work with, and do not know ow to program, let me know as soon as possible, and we will find an alternative.

(As extra credit problem, write a program for Francis’s algorithm as well, together with an explanation of your code, and apply the algorithm to the same 4\times 4 matrix.)

For the remaining problems, you should turn in your own work. You can still collaborate with others, but please make sure to give appropriate credit and indicate clearly who you worked with, what references you consulted, etc:

  • Give an example of a matrix for which the power method fails. (Include a proof that this is indeed the case.)
  • Let A be an n\times n matrix with complex entries. Consider it as a linear transformation A:\mathbb C^n\to\mathbb C^n. Verify explicitly that A^* is the unique matrix with the property that u\cdot A^*v=(Au)\cdot v for all vectors u,v. Recall that for u=(u_1,\dots,u_n)^T and v=(v_1,\dots,v_n)^T, their dot product u\cdot v is given by u\cdot v=u_1\bar{v_1}+u_2\bar{v_2}+\dots+u_n\bar{v_n}.
  • Check that if A=A^*, then the eigenvalues of A are real. To see this, let u be an eigenvector of A, and consider (Au)\cdot u. In particular, this means that the eigenvalues of a symmetric matrix are real.

More significantly, for any Hermitian matrix A, that is, one that satisfies A=A^*, there is a basis consisting of eigenvectors of A. We will prove this in due time.

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